Graphs

1
Graphs Linear Equations
Y
ymxb
X
2
Example of a Linear Function
Y
77

(11,77)
A Dogs Humans Equivalent Age
yf(x)7x

35
(5,35)

21
(3,21)
X
(0,0)
3 5 11
A Dogs Actual Age
3
Major Elements of Graphing Lines
Graphing Ordered Pairs Graphing
Equations Linear Equations Slope
Equations Finding Equations of Lines Fitting
Equations to Lines Parallel Perpendicular
Lines
4
A Point
Y
(X,Y) (4,3)
X
(X,Y) is called an Ordered Pair The X value or X
Coordinate is the location of a point in the X
direction The Y value or Y Coordinate is the
location of a point in the Y direction
5
How to Graph a Point
Y
(X,Y) (4,2)
321
X
1 2 3 4
-4-3-2-1 0
X is the distance along the xaxis Y is the
distance along the yaxis
Y
3210
HINT Think of the x-axis as the Number Line
X
-1
1 2 3 4
-4-3-2-1 0
-2
HINT Think of the y-axis a vertical Number Line
6
Important Vocabulary for Graphs
The Graph itself is called the x-y plane (ie.
Plane surface) or The Coordinate Plane or
Cartesian Coordinate Plane after Renee Descartes
Y
y-axis
1st Quadrant
2nd Quadrant
(X,Y)
(-X,Y)
X
x-axis
(X,-Y)
(-X,-Y)
3rd Quadrant
4th Quadrant
Quadrants start with Positive X Y and go
Counter Clockwise
7
Graphing Linear Equations(Find 3 Domain Range
Points)
First Degree Equations are Lines (ymxb) and you
calculate 3 (X,Y) values Make sure the points
line up on a x-y graph and connect the dots.
8
RECALL X-Domain Y-Range
Graphing Lines is just like finding the Range of
3 Domain Points(Substitute each Domain value
into the equation)

• y 2x-7 when the Domain is -2, 0, 2
• f(-2) 2(-2) -7 -4 -7 -11 (-2,-11)
• f(0) 2(0) -7 0 -7 -7 (0,-7)
• f(2) 2(2) -7 4 -7 -3 (2,-3)

Answer RANGE -11, -7, -3
9
Practice Finding 3 PointsGiven a Linear Equation
Find any 3 (X,Y) points for the following
equations y5x y4x-5 y3x1 (Hint Try x0)
10
Sample Solutions

• x y 5x
• 0 0
• 5
• 10

x y 4x-5
x y 3x1
11
Now Graph the 3 Points

• x y 5x
• 0 0
• 5
• 10

(2,10)

(1,5)

(0,0)
12
What is Intercept in Math?
Y
X
An Intercept is the coordinate where a line
crosses the x or y axis
13
Using XY Intercepts to Graph a Line
Y
y-axis
The Y intercept is the y coordinate (where a line
crosses the y axis).

(0,2)
(3,0)

X
x-axis
The X intercept is the x coordinate (where a line
crosses the x axis).
14
Name the XY Intercepts
Y
y-axis

(0,2)
(3,0)

X
x-axis
15
Name the XY Intercepts
Y
y-axis

(0,2)

X
x-axis
(-2,0)
16
Name the XY Intercepts
Y
y-axis
X
x-axis
17
Name the XY Intercepts
Y
y-axis
X
x-axis
18
What is the value of x at the y intercept? What
is the value of Y at the x-intercept?
Y
y-axis
X
x-axis
19
Graph y 2x - 6 using xy intercepts
Graph Linear Eq.

• X Y 2x - 6

1st Make x-y table
Y
y-axis
X
x-axis
20
Graph y 2x - 6 using xy intercepts
Graph Linear Eq.

• X Y 2x - 6
• 0 -6

1st Make x-y table 2nd Set x 0 and solve for
y
Y
y-axis
X
x-axis
(0,-6)
21
Graph y 2x - 6 using xy intercepts
Graph Linear Eq.

• X Y 2x - 6
• 0 -6
• 0

1st Make x-y table 2nd Set x 0 and solve for
y 3rd Set y 0 and solve for x
Y
y-axis
X
(3,0)
x-axis
(0,-6)
22
Graph y 2x - 6 using xy intercepts
Graph Linear Eq.

• X Y 2x - 6
• 0 -6
• 0

1st Make x-y table 2nd Set x 0 and solve for
y 3rd Set y 0 and solve for x 4th Plot these 2
points and draw line
Y
y-axis
X
(3,0)
x-axis
(0,-6)
23
Graph y 2x - 6 using xy intercepts
Graph Linear Eq.

• X Y 2x - 6
• 0 -6
• 0
• 4 2

1st Make x-y table 2nd Set x 0 and solve for
y 3rd Set y 0 and solve for x 4th Plot these 2
points and draw line 5th Use 3rd point to check
Y
y-axis
(4,2)
X
(3,0)
x-axis
(0,-6)
24
Graphing Horizontal Vertical Lines
This line has a y value of 4 for any x-value.
Its equation is y 4 (meaning y always equals
4)
Y
y-axis
X
x-axis
25
Graphing Horizontal Vertical Lines
This line has a x value of 1 for any y-value.
Its equation is x 1 (meaning x always equals
1)
Y
y-axis
X
x-axis
26
The Equation of a Vertical Line is XConstant
Y
y-axis
x 1
X
x-axis
27
The Equation of a Horizontal Line is YConstant
Y
y-axis
y 3
X
x-axis
28
Graph the following lines
Y -4 Y 2 X 5 X -5 X 0 Y 0
29
Answers
x -5
x 5
Y
y-axis
X
x-axis
30
Answers
Y
y-axis
y 2
X
x-axis
y -4
31
Answers
Y
y-axis
y 0
X
x-axis
x 0
32
SLOPE
NEGATIVE-DOWN
POSITIVE-UP
Slope is a measure of STEEPNESS
33
The Symbol forSLOPE m
NEG. Slope is -m
POS. Slope is m
Think of m for Mountain
34
SLOPE
(6,4)

4
3
(3,2)
2

1
(0,0)
2
1
3
4
5
6
How much does this line rise? How much does
it run?
35
SLOPE
2
3
(6,4)

4
3
(3,2)
2

1
(0,0)
2
1
3
4
5
6
How much does this line rise? How much does
it run?
2
3
36
mSLOPE
x2y2
(6,4)

4
x1y1
3
(3,2)
2

1
(0,0)
2
1
3
4
5
6
37
Switch points and calculate slopeMake (3,2)
(x2,y2) (6,4) (x1,y1)
(x2,y2)(64)
(x1,y1)(6,4)


(x1,y1)(3,2)
(x2,y2)(3,2)


38
Recalculation with points switched
(x1,y1)(6,4)

(x2,y2)(3,2)

Same slope as before
39
It doesnt matter what 2 points you choose on a
linethe slope must come out the same
40
Keeping Track of Signs When Finding The Slope
Between 2 Points
Be Neat Careful Use (PARENTHASES) Double
Check Your Work as you Go Follow 3 Steps
41
3 Steps for finding the Slope of a line between 2
Points(3,4)(-2,6)
1st Step Write x1,y1,x2,y2 over numbers 2nd
Step Write Formula and Substitute x1,x2,y1,y2
values. 3rd Step Calculate Simplify
x1 y1
x2 y2
(3,4) (-2,6)
42
Find the Slopes of Lines containing these 2 Points
1. (1,7) (5,2)
2. (3,5) (-2,-8)
3. (-3,-1) (-5,-9)
4. (4,-2) (-5,4)
5. (3,6) (5,-5)
6. (1,-4) (5,9)
43
ANSWERS
1. (1,7) (5,2)
2. (3,5) (-2,-8)
3. (-3,-1) (-5,-9)
4. (4,-2) (-5,4)
5. (3,6) (5,-5)
6. (1,-4) (5,9)
44
Solve for y if (9,y) (-6,3) m2/3
45
Review Finding the Slopes of Lines Given 2 Points
1st Step Write x1,x2,y1,y2 over numbers 2nd
Step Write Formula and Substitute x1,x2,y1,y2
values. 3rd Step Calculate Simplify
NOTE Be Neat, Careful, and Precise and Check
your work as you go..
46
Negative Slope Is Down the Hill
Positive Slope Is Up the Hill
NO Slope Vertical Drop
ZERO Slope Horizontal
47
NO Slope Vertical Drop
ZERO Slope Horizontal
48
Equations of a Line
There are 3 Forms of Line Equations Standard
Form axbyc Slope Intercept
Form ymxb Point-Slope Form y-y1m(x-x1) All
3 describe the line completely but are used for
different purposes. You can convert from one form
to another.
49
Converting fromStandard Form axbycto Slope
Intercept Form
JUST SOLVE FOR Y
Slope Intercept Form ymxb
50
Slope Intercept Form ymxb
The great thing about this form is b is the
y-intercept. This makes graphing a line
incredibly easy. Check it out. If
The y intercept is 1
(0,1)

Almost a free point on graph
51
Slope Intercept Form ymxb
All you have to do now is use the slope to rise
and run from the intercept connect the points.

(0,1)

Rise 2 and Run 3 from the y-intercept connect
points.
52
ymxb when m is negative
All you have to do now is use the slope to rise
and run from the intercept connect the points.
(0,1)


Rise -2 and Run 3 from the y-intercept connect
points.
53
Slope Intercept Form ymxbGRAPH THESE LINEAR
EQUATIONS
Label y-intercept Use one big graph
54
If linear equation is not in ymxb form solve
for y
Solution Steps to Solve for y Divide by 2
Now it is This line has an y intercept of -2 and
rises 5 and runs 2.
55
Graphing a line with slope intercept equation
Graph

• 1. Solve for y
• Y-Intercept is 1st Point.
• From the y-intercept
• Rise 5 and run 2 for
• Second Point.
• 4. Connect Points with line.

2

5
(0,-2)
56
Now it is easy to graph


(0,-2)
57
Put into slope-intercept form and graph
58
Review Steps of Graphing from the Slope Intercept
Equation

1. Make sure equation is in ymxb form
2. Plot b(y-intercept) on graph (0,b)
3. From b, Rise and Run according to the slope to
   plot 2nd point.
4. Check sign of slope visually

NEG. Slope is -m
POS. Slope is m
59
Find the Equation of a Line(Given Pt. Slope)
Given a point (2,5) m5 Write the Equation

1. Write Slope-Intercept Equation
2. 2. Plug-in (x,y) m values
3. Solve for b
4. Plug m b into Slope-Int. Eq.

60
Find the Equation of a Line(Given Pt. Slope)
Method 2Using the Pt.-Slope Eq.
Given a point (2,5) m5 Write the Equation

1. Write Pt.-Slope Equation
2. 2. Plug-in (x,y) m values
3. Solve for y

61
Find the Equation of a Line(Given 2 Points)
Given a point (x1,y1) (x2,y2) (2,5) (3,10)

1. Find Slope using
2. Write Slope-Intercept Equation
3. Plug-in (x,y) m values
4. Solve for b
5. Plug m b into Slope-Int. Eq.

62
Parallel LinesHave the Same Slope
5

4
3
2

1
(0,0)
2
1
3
4
5
6
63
Perpendicular LinesHave Neg. Reciprocal Slopes
3
2
1
(0,0)
2
1
3
4
5
6
64
Systems of Equations
Given 2 linear equations The single point where
they intersect is a solution to either
equation It is also the solution to both
equations or what we call the solution to the
SYSTEM OF EQUATIONS

65
(0,3)
3
2
Solution
(2,1)
1
(0,0)
2
1
3
4
5
6
-1
(0,-1)
66
(0,3)
3
2
Solution
(2,1)
1
(0,0)
2
1
3
4
5
6
-1
(0,-1)
67
Systems of Equations
The Solution is where the two lines meet (or
intersect)
(0,3)
3
2
Solution
(2,1)
1
(0,0)
2
1
3
4
5
6
-1
(0,-1)